Moment of inertia of a ring about an axis passing through its center and perpendicular to its plane : Suppose a ring of mass M and radius R is placed in X- Y plane with its center at the origin of cartesian system. Z-axis will be our required axis in the question.
The mass of the ring is uniformly distributed to its whole circumference. Therefore, its whole circumference can be supposed to be made up of n parts of masses respectively m1, m2, m3, …………., mn.
Each mass element is situated at the same distance R from the Z-axis (i.e, schedued axis). Therefore the moment inertia of the ring,
I = I1 + I2 + I3 + ……. +In
= m1R2 + m2R2 + m3R2 + …………… + mnR2
= (m1 + m2 + m3 + ……….. + mn )R2
or I = MR2
where M = mass of ring.